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4. Trigonometry on the Calculator.
We note that the calculator has keys labeled SIN, COS, and TAN. These are, as might be expected, the keys for the sine, the cosine, and the tangent functions respectively. If the real number x is on level 1 of the stack, pressing SIN will give sin(x), TAN will give tan(x), and COS will give cos(x). The real number x will be interpreted as degrees or radians according to the angle mode setting. For example if the number 30 is on level 1 of the stack and the calculator is in degree mode, pressing SIN gives the result .5 as expected. If, however, the calculator is in radian mode with 30 on level 1 of the stack, the result of pressing SIN is -.988031624093, which is the sine of 30 radians.
There are no keys for the cosecant, secant, or cotangent functions. Since these functions are the reciprocals of the sine, cosine, and tangent respectively, they can be obtained with the keys we have and the 1/x key. The trigonometric functions are on pages 3 - 6 of UG.
Calculator Example 3.4.1
Find 
Solution: With the calculator in radian mode key in LS 
COS 1/x. We see the result 2.00000000001. The answer, of
course, should be 2, but we are again seeing an example of the round off errors caused by using a finite machine to
approximate computations with real numbers.
We see that the left shift functions for SIN, COS, and TAN are labeled ASIN, ACOS, and ATAN, respectively. These are, respectively, the Arcsine, Arccosine, and Arctangent functions. These functions are not inverses of each other since, for example, SIN is the sine function, not the Sine function.
then take the Arctangent of the result.
TAN, and we see the expected answer -1. Now key in
LS ATAN and we get the result -.785398163397, which is the decimal approximation for 
This should not be
surprising since 
is not in the domain of Tan and so 
is not necessarily 
for this angle.
on the calculator.
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Calculator Example 3.4.4 Graph y = Cos(x) and y = Arccos(x) on the same axes. See Problem 7, Exercises for Chapter III Section 2. Solution: Assuming your calculator is as you left it from the
previous example, press CANCEL or F6-CANCL to return
to the PLOT WINDOW dialog box. Now key NXT F1-RESET DA F6-OK. This resets the plot parameters back to
their default state. In this state one unit in both directions is
10 pixels, so geometric properties will not be distorted, but
the two views are about twice as big as we
need them. We will cut each of the fields in the two view areas in half, and will add .5 to the two V-VIEW fields to move
the origin down that amount. To accomplish this, press F2-CALC 2 Now press 0 ENTER so the graph will start at x = 0 and LS
Now press NXT F6-OK LS(hold) 2D/3D to get into the PLOT SETUP dialog box. Key RA +/- to change the angle mode to radians, then RA ' COS F1-X ENTER to change to the cosine function. Now key DA .5 ENTER .5 ENTER to change the ticks to .5 units in both directions. The screen should now look like Figure 4b. Now press F5-ERASE F6-DRAW, and when the graph is complete, press the minus sign to remove the menu. You should see the graph shown in Figure 4c. Press CANCEL to get back to the PLOT SETUP dialog box and change the EQ: field to ACOS(X). Now press LS(hold) WIN to get into the PLOT WINDOW and change Indep Low: to -1 and High: to 1 to plot the domain [-1, 1]. Press F6-DRAW without erasing. You should now see the graph shown in Figure 4d. Finally, let us add the line of symmetry for cos(x) and Arccos(x).
Press F2-(X,Y) and use RA and UA to move the cursor
until the coordinates at the bottom of the screen show X:2. Y:2.
(NOTE: holding the arrow keys down will cause the
cursor to move more quickly.)
Now press |
|
 Figure 4e |
 Figure 4f |
F6-OK to set the left field of H-VIEW. Now key
NXT F2-CALC 2
F6-OK to set the right field of H-VIEW. For each of the two fields of V-VIEW key NXT F2-CALC
2
.5 + F6-OK. 
to end the graph at x = 
Your screen should now look
like Figure 4a.
to put a mark on the screen as shown in Figure 4e. Next press + to get the
menu back, then press F5-EDIT. Use DA and LA to move the cursor back to the origin, press F3-LINE, then 
and you
should see the graph shown in Figure 4f.
Calculator Example 3.4.5
In 
and c = 47.26. Solve the triangle. Find the angles to one decimal place and the lengths
to two decimal places.
Solution: First 
so on the calculator we key 180 ENTER 37.1 - 58.3 - and we see that 
From the Law of Sines, 
We solve this for a and substitute the known quantities to get
With the calculator in degree mode and set to Fix 2, we key in 47.26 ENTER 37.1 SIN 
84.6 SIN
and obtain a = 28.63. Similarly
Calculator Example 3.4.6
Given that in 
one has 
a = 22.16, and b = 43.26, solve the triangle. Give angles to three decimal places and
lengths to 2 decimal places.
Solution: We observe that the given information is of the form SAS, hence the triangle is determined. We first use the Law of Cosines to find
On the calculator (in radian mode) this is accomplished with 22.16 LS x2 43.26 LS x2 + 2 ENTER 22.16 
43.26
COS 
which gives us c = 26.50. We now use the Law of Sines to find
or
Before we continue, a reminder. The SWAP command can be found by TOOL F3-STACK F2-SWAP. It can also be found by LS RA, and if the calculator is in a state where RA would not make sense, the LS is not necessary.
Set the calculator to Fix 3. Now, assuming the value of c is still on the stack from the previous calculation, we proceed with
22.16 ENTER LS 
SIN 
LS ASIN, which gives us 
( Notice that when we were ready to divide
by c we took advantage of the fact that it was already on the stack and just used the SWAP command to put it in the right
position for the division. Every time one can eliminate the need to key in a number, one has removed an opportunity to
make an error.) Finally, we compute the last angle by subtracting the first two from 
and get 
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